Crossing minimization in linear embeddings of graphs

Masuda, Seizo; Nakajima, Kazuo; Kashiwabara, Toshinobu; Fujisawa, Toshio

doi:10.1109/12.46286

Cited by 57 publications

(41 citation statements)

References 10 publications

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“…We use a reduction from Thereby, an f -linear drawing is one where all vertices are placed on a horizontal line, each vertex v at coordinate f (v), and each edge is either drawn completely above or completely below that line. It was shown in [23] that Flcn is NPcomplete.…”

Section: A Ziegler's Proof Of Np-hardness Of Meimentioning

confidence: 99%

A tighter insertion-based approximation of the crossing number

Chimani

Hliněný

2016

J Comb Optim

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Abstract. Let G be a planar graph and F a set of additional edges not yet in G. The multiple edge insertion problem (MEI) asks for a drawing of G + F with the minimum number of pairwise edge crossings, such that the subdrawing of G is plane. Finding an exact solution to MEI is NP-hard for general F . We present the first polynomial time algorithm for MEI that achieves an additive approximation guarantee -depending only on the size of F and the maximum degree of G, in the case of connected G. Our algorithm seems to be the first directly implementable one in that realm, too, next to the single edge insertion. It is also known that an (even approximate) solution to the MEI problem would approximate the crossing number of the F -almost-planar graph G+F , while computing the crossing number of G+F exactly is NP-hard already when |F | = 1. Hence our algorithm induces new, improved approximation bounds for the crossing number problem of F -almost-planar graphs, achieving constant-factor approximation for the large class of such graphs of bounded degrees and bounded size of F .

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Section: A Ziegler's Proof Of Np-hardness Of Meimentioning

confidence: 99%

A tighter insertion-based approximation of the crossing number

Chimani

Hliněný

2016

J Comb Optim

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“…, |V |}, and an integer Z, is there a drawing of G with the vertices placed on a horizontal line (spine) in the order specified by f and the edges drawn as semicircles above or below the spine so that there are at most Z crossings? Masuda et al [8] showed that the problem is NP-complete, even if G is a matching.…”

Section: Theorem 8 Given a Set Of N Input Points In K Different Colomentioning

confidence: 99%

“…The proof is by reduction from the NP-complete Fixed Linear Crossing Number problem [8]: Given a graph G = (V, E), a bijective function f : V → {1, . .…”

Section: Theorem 8 Given a Set Of N Input Points In K Different Colomentioning

confidence: 99%

Many-to-One Boundary Labeling with Backbones

et al. 2013

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Abstract. In this paper we study many-to-one boundary labeling with backbone leaders. In this model, a horizontal backbone reaches out of each label into the feature-enclosing rectangle. Feature points associated with this label are linked via vertical line segments to the backbone. We present algorithms for label number and leader-length minimization. If crossings are allowed, we aim to minimize their number. This can be achieved efficiently in the case of fixed label order. We show that the corresponding problem in the case of flexible label order is NP-hard.

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“…Due to the complexity of the crossing minimization problem, many restricted versions have been considered in the literature. However, in most cases, e.g., for bipartite, linear, and circular drawings, the problem remains NP-hard [6,16,15].…”

Section: Introductionmentioning

confidence: 99%

Exact Crossing Minimization

et al. 2006

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Abstract. The crossing number of a graph is the minimum number of edge crossings in any drawing of the graph into the plane. This very basic property has been studied extensively in the literature from a theoretic point of view and many bounds exist for a variety of graph classes. In this paper, we present the first algorithm able to compute the crossing number of general sparse graphs of moderate size and present computational results on a popular benchmark set of graphs. The approach uses a new integer linear programming formulation of the problem combined with strong heuristics and problem reduction techniques. This enables us to compute the crossing number for 91 percent of all graphs on up to 40 nodes in the benchmark set within a time limit of five minutes per graph.

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Crossing minimization in linear embeddings of graphs

Cited by 57 publications

References 10 publications

A tighter insertion-based approximation of the crossing number

A tighter insertion-based approximation of the crossing number

Many-to-One Boundary Labeling with Backbones

Exact Crossing Minimization

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